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About adjoint and \((h,j)\)-adjoint digraphs of \(k\)-regular multigraphs. (Sobre digrafos adjuntos y \((h,j)\) adjuntos de multidigrafos \(k\)-regulares.) (Spanish. English summary) Zbl 1099.05507

Summary: This work connects graph theory with matrix theory. We prove that every \({}^{(h,j)}G\) digraph of a \(k\)-regular multidigraph on \(n\) vertices has exactly \([k^{(h-j)}!]^{n\cdot k^j}\) different covering \((k^{(h-j)}-1)\)-regular subdigraphs. The proof is via a suitable matrix representation, using the permanent of the precedence matrix of the \((h,j)\)-adjoint digraphs.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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